Can a NN be configured to indicate which points of the input influenced its prediction and how?
Yes there definitely is, and research into this has actually resulted in some really cool behaviour.

One of the simplest ways is to simply back propagate the gradient all the way back to the input. Areas of the input that affected the final decision will receive larger gradients. Interestingly, this also sort of works as a rudimentary form of semantic segmentation.

Other ways are to change segments of the input and to see how that affects the output (like add 0.1 to a pixel).

You can also determine what each filter in a convolutional layer is looking at using similar techniques.

It's a super interesting field of machine learning, and I highly recommend taking a look at this lecture that is completely free and one of the most interesting ones I've personally seen. it will explain all this much better than I have done.